## Engineering Maths 3 - Dec 2015

### Computer Engg (Semester 4)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.

### Solve any one question from Q1 and Q2

**1 (a) (i)** (D^{2}-2D-3) y=3e^{-3x} sin e^{-3x} + cos (e^{-3x}).(4 marks)
**1 (a) (ii)** (D^{2}-2D+2) y=e^{x} tan x. (By variation of parameters).(4 marks)
**1 (a) (iii)** $$ x^2 \dfrac {d^2 y}{dx^2}-2x \dfrac {dy}{dx} - 4y=x^2 $$(4 marks)
**1 (b)** Find the Fourier transform of e^{-|x|} and hence show that: $$ \int^{\infty}_{-\infty} \dfrac {e^{i\lambda x}}{1+\lambda^2} d \lambda = \pi e^{-|x|} $$(4 marks)
**2 (a)** An unchanged condenser of capacity C charged by applying an e.m.f. of value $ \dfrac {t}{\sqrt{LC}} $ through the LEDs of inductance L and of negligible resistance. The charge Q on the place of condenser satisfied the differential equation: $$ \dfrac {d^2Q}{dt^2} + \dfrac {Q}{LC} = \dfrac {E}{L}\sin \dfrac {t}{\sqrt{LC}} $$ Prove that the charge at any time t is given by: $$ Q= \dfrac {EC}{2} \left [ \sin \dfrac {t}{\sqrt{LC}} - \dfrac {t}{\sqrt{LC}} \cos \dfrac {t}{\sqrt{LC}} \right ] $$(4 marks)
**2 (b)** Find the Inverse Z-transform (any one): $$ i) \ F(z) = \dfrac {z+2}{z^2 - 2z+1} \ \text {for }|z|>1. \\
ii) \ F(z) = \dfrac {10z}{(z-1)(z-2)} \ \text {(Use inversion integral method).}$$(4 marks)
**2 (c)** Solve the following difference equation to find {f(k)}: $$ f(k+1)+ \dfrac {1}{4} f(k) = \left ( \dfrac {1}{4} \right )^k, \ k\ge 0, \ f(0)=0 $$(4 marks)

### Solve any one question from Q3 and Q4

**3 (a)** The first four moments of distribution about the value 4 are -1.5, 17, -30 and 108. Obtain the first four central moments, mean, standard deviation and coefficient of skewness and kurtosis.(4 marks)
**3 (b)** If the probability that a concrete cube fails is 0.001. Determine the probability that out of 1000 cubes:

i) exactly two

ii) more than one cubes will fail.(4 marks)
**3 (c)** Show that: $$\overline F= ( y \sin z - \sin x) \overline i (x \sin z +2 yz)\overline j + (xy\cos z+y^2)\overline k $$ is irrotational and hence find scalar function ϕ s.t. F=∇ϕ.(4 marks)
**4 (a)** Find the directional derivative of ϕ=4xz^{3}-3x^{2}y^{2}z at (2, -1, 2) along a line equally inclined with co-ordinate axes.(4 marks)
**4 (b)** For a solenoidal vector field F, show that:

curl curl curl curl F=∇^{4}F.(4 marks)
**4 (c)** The regression equations are:

8x+10y+66=0 and 40x-18y=214.

The value of variance of x is 9. Find.

i) The mean values of x and y

ii) The correlation coefficient between x and y

iii) The standard deviation of y.(4 marks)

### Solve any one question from Q5 and Q6

**5 (a)** Find the work done in moving a particle once round the ellipse: $$ \dfrac {x^2}{16}+ \dfrac {y^2}{4}=1, \ z=0 $$ under the field of force given by: $$ \overline{F} = (2x-y+z)\overline i + (x+y-z^2) \overline j + (3x-2y + 4z) \overline{k} $$(4 marks)
**5 (b)** Evaluate: $$ \iint_s (\nabla \times \overline{F} ) \cdot \widehat{n} \ dS $$ where $$ \overline F = (x^3 - y^3) \overline {i} - xyz \overline j + y^2 \overline {k} $$ and S is the surface x^{2}+4y^{2}+z^{2}-2x=4 above the plane x=0.(4 marks)
**5 (c)** Evaluate: $$ \iint_s \overline {F} \cdot \overline {dS} $$ using divergence theorem, where $$ \overline F= x^3 \overline i + y^3 \overline j+z^3 \overline k $$ and S is the surface of sphere x^{2}+y^{2}+z^{2}=a^{2}.(5 marks)
**6 (a)** If $ \overline F = x^2 \overline i + (x-y) \overline j + (y+z) \overline k $ displaces a particle from A(1, 0, 1) to B(2, 1, 2) along the straight line AB, find work done.(4 marks)
**6 (b)** Evaluate: $$ \int_C (e^x dx + 2ydy - dx) $$ where C is the curve x^{2}+y^{2}=4, z=2.(4 marks)
**6 (c)** Evaluate: $$ \int_s \overline F \cdot \overline {dS} $$ using Gauss divergence theorem, where: $$ \overline F = 2xy\overline i + yz^2 \overline j + xz\overline k $$ and S is the region bounded by:

x=0, y=0, z=0, y=3, x+2z=6.(5 marks)

### Solve any one question from Q7 and Q8

**7 (a)** Show that u=y^{3}-3x^{2}y is harmonic function. Find its harmonic conjugate and the corresponding analytic function f(z) in terms of z.(5 marks)
**7 (b)** Using Cauchy's integral formula, evaluate: $$ \int_C \dfrac {2z^2 + z +5}{(z-3 /2)^2}dz $$ where C is $ \dfrac {x^2}{4} + \dfrac {y^2}{9} = 1. $(4 marks)
**7 (c)** Find the bilinear transformation which maps the points z=1, i, -1, onto the points w=0, 1, ∞.(4 marks)
**8 (a)** If f(z) is an analytic function v^{2}=u, then show that f(z) is constant function.(4 marks)
**8 (b)** Using residue theorem evaluate: $$ \int_C \dfrac {z}{z^4 +13z^2 + 36} dz $$ where 'C' is the circle $ z| = \dfrac {5}{2}. $(5 marks)
**8 (c)** Find the map of the circle |z=i|=1 under the transformation $ w=\dfrac {1}{w} $ into w-plane.(4 marks)